By a result of Kostant [1]}, the multiplication map gives a bijection from \(W(\mathfrak {g}, \mathfrak {t})^1\times W(\mathfrak {k}, \mathfrak {t}_f)\)  onto \(W(\mathfrak {g}, \mathfrak {t}_f)\) . In particular, this explains that
\(63=|W(\mathfrak {g}, \mathfrak {t}_f)^1|=\frac{|W(\mathfrak {g}, \mathfrak {t}_f)|}{|W(\mathfrak {k}, \mathfrak {t}_f)|}=\frac{2903040}{46080}.\) 
